数学最新文献

{"title":"Generalized co-polynomials of RII type and associated quadrature rules","authors":"Vinay Shukla ,&nbsp;A. Swaminathan","doi":"10.1016/j.cam.2025.116957","DOIUrl":"10.1016/j.cam.2025.116957","url":null,"abstract":"<div><div>When the co-recursion and co-dilation in the recurrence relation of certain sequences of orthogonal polynomials are not at the same level, the behavior of the modified orthogonal polynomials is expected to have different properties compared to the situation of the same level of perturbation. This manuscript attempts to derive structural relations between the perturbed and original <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>I</mi><mi>I</mi></mrow></msub></math></span> type orthogonal polynomials. The classical result is improved using a transfer matrix approach. It turns out that the <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>I</mi><mi>I</mi></mrow></msub></math></span> fraction with perturbation is the rational spectral transformation of the unperturbed one. The derived notions are used to deduce some consequences for the polynomials orthogonal on the real line. A natural question that arises while dealing with perturbations at different levels, i.e., which perturbation, co-recursion or co-dilation, needs to be performed first, is answered.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"474 ","pages":"Article 116957"},"PeriodicalIF":2.6,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144748545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Existence and uniqueness of time-periodic solutions to the Oberbeck–Boussinesq system","authors":"Tomoyuki Nakatsuka","doi":"10.1016/j.nonrwa.2025.104461","DOIUrl":"10.1016/j.nonrwa.2025.104461","url":null,"abstract":"<div><div>This paper is devoted to the study of the time-periodic problem for the Oberbeck–Boussinesq system in the whole space. Our investigation is based on the reformulation of the time-periodic problem and does not depend on the analysis of the initial value problem. We construct a time-periodic solution with more information on its structure than the solutions in preceding studies. We also prove that our solution, small in an appropriate sense, is unique in the class of solutions having slightly more regularity.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"88 ","pages":"Article 104461"},"PeriodicalIF":1.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722113","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Continuum limit of fourth-order Schrödinger equations on the lattice","authors":"Jiawei Cheng,&nbsp;Bobo Hua","doi":"10.1112/jlms.70247","DOIUrl":"https://doi.org/10.1112/jlms.70247","url":null,"abstract":"<p>In this paper, we consider the discrete fourth-order Schrödinger equation on the lattice <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$hmathbb {Z}^2$</annotation>\u0000 </semantics></math>. Uniform Strichartz estimates are established by analyzing frequency localized oscillatory integrals with the method of stationary phase and applying Littlewood–Paley inequalities. As an application, we obtain the precise rate of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$L^2$</annotation>\u0000 </semantics></math> convergence from the solutions of discrete semilinear equations to those of the corresponding equations on the Euclidean plane <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math> in the continuum limit <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>h</mi>\u0000 <mo>→</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$h rightarrow 0$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory","authors":"David Fernández-Bretón,&nbsp;Jareb Navarro-Castillo,&nbsp;Jesús A. Soria-Rojas","doi":"10.1112/jlms.70249","DOIUrl":"https://doi.org/10.1112/jlms.70249","url":null,"abstract":"&lt;p&gt;We study ultrafilters from the perspective of the algebra in the Čech–Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a Q-point (resp., a selective ultrafilter) and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal F^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (resp., &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal G^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) is the smallest family containing &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and closed under iterated sums (resp., closed under Blass–Frolík sums and Rudin–Keisler images), then &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;F&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal F^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; (resp., &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;G&lt;/mi&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathcal G^p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;mo&gt;[&lt;/mo&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;mo&gt;]&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathnormal {mathbf {L}(mathbb {R})}[p]$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; where &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;L&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathnormal {mathbf {L}(mathbb {R})}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a Solovay model (of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ZF&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathnormal {mathsf {ZF}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; without choice) and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;p&lt;/mi&gt;\u0000 &lt;annotation&gt;$p$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a selective ultrafilter, there are no idempotent elements. In particular, the theory &lt;sp","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Perfect k-matching, k-factor-critical and Aα-spectral radius","authors":"Mengyuan Niu ,&nbsp;Shanshan Zhang ,&nbsp;Xiumei Wang","doi":"10.1016/j.dam.2025.07.020","DOIUrl":"10.1016/j.dam.2025.07.020","url":null,"abstract":"<div><div>A <span><math><mi>k</mi></math></span>-<em>matching</em> of a graph <span><math><mi>G</mi></math></span> is a function <span><math><mi>f</mi></math></span>: <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math></span> satisfying <span><math><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>∂</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></munder><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>≤</mo><mi>k</mi></mrow></math></span> for any vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. A <span><math><mi>k</mi></math></span>-matching of a graph <span><math><mi>G</mi></math></span> is <em>perfect</em> if <span><math><mrow><munder><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>∂</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></munder><mi>f</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>k</mi></mrow></math></span> for every vertex <span><math><mrow><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. A graph <span><math><mi>G</mi></math></span> of order <span><math><mi>n</mi></math></span> is <em>k-factor-critical</em> if the removal of any set of <span><math><mi>k</mi></math></span> vertices of <span><math><mi>G</mi></math></span> results in a graph with a perfect matching. Let <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the adjacency matrix and the degree diagonal matrix of <span><math><mi>G</mi></math></span>. For <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, Nikiforov <span><math><mrow><mo>(</mo><mn>2017</mn><mo>)</mo></mrow></math></span> introduced the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-matrix of G as follows: <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>.</mo></mrow></math></span> In this paper, according to the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius, we provide two sufficient conditions to ensure that a graph is <span><math><mi>k</mi></math></span>-factor-critical and has a perfect <span><math><mi>k</mi></math></span>-matching, respectively.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"376 ","pages":"Pages 384-393"},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Indivisibility for classes of graphs","authors":"Vincent Guingona ,&nbsp;Felix Nusbaum ,&nbsp;Zain Padamsee ,&nbsp;Miriam Parnes ,&nbsp;Christian Pippin ,&nbsp;Ava Zinman","doi":"10.1016/j.disc.2025.114703","DOIUrl":"10.1016/j.disc.2025.114703","url":null,"abstract":"<div><div>We examine indivisibility for classes of graphs. We show that the class of hereditarily <em>α</em>-sparse graphs is indivisible if and only if <span><math><mi>α</mi><mo>&gt;</mo><mn>2</mn></math></span>. Additionally, we show that the following classes of graphs are indivisible: perfect graphs, cographs, and chordal graphs, and the following classes of graphs are not indivisible: threshold graphs, split graphs, and distance-hereditary graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114703"},"PeriodicalIF":0.7,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144721907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the complexity of p-order cone programs","authors":"Víctor Blanco ,&nbsp;Victor Magron ,&nbsp;Miguel Martínez-Antón","doi":"10.1016/j.jco.2025.101979","DOIUrl":"10.1016/j.jco.2025.101979","url":null,"abstract":"<div><div>This manuscript explores novel complexity results for the feasibility problem over <em>p</em>-order cones, extending the foundational work of Porkolab and Khachiyan (1997) <span><span>[30]</span></span>. By leveraging the intrinsic structure of <em>p</em>-order cones, we derive refined complexity bounds that surpass those obtained via standard semidefinite programming reformulations. Our analysis not only improves theoretical bounds but also provides practical insights into the computational efficiency of solving such problems. In addition to establishing complexity results, we derive explicit bounds for solutions when the feasibility problem admits one. For infeasible instances, we analyze their discrepancy quantifying the degree of infeasibility. Finally, we examine specific cases of interest, highlighting scenarios where the geometry of <em>p</em>-order cones or problem structure yields further computational simplifications. These findings contribute to both the theoretical understanding and practical tractability of optimization problems involving <em>p</em>-order cones.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"91 ","pages":"Article 101979"},"PeriodicalIF":1.8,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Distributed robust anti-interference Nash equilibrium search for second-order multi-agent systems and its application","authors":"Yong Chen ,&nbsp;Bowen Hao ,&nbsp;Fuxi Niu ,&nbsp;Jiarui Li ,&nbsp;Xiaohong Nian ,&nbsp;Hailiang Hou","doi":"10.1016/j.matcom.2025.07.042","DOIUrl":"10.1016/j.matcom.2025.07.042","url":null,"abstract":"<div><div>This paper focuses on achieving a distributed Nash equilibrium while accounting for both matched and mismatched disturbances. It introduces an algorithm that relies on disturbance estimation and a distributed feedforward controller. The algorithm is mainly divided into two parts. The first part is to design effective disturbance observers for different types of disturbances, and the second part is to design distributed feedforward feedback control algorithm utilizing these disturbance observations. By employing this algorithm, the agents’ states ultimately converge to Nash equilibrium points. The convergence of the algorithm is proved by the input state stability criterion and Lyapunov stability analysis. Finally, a numerical simulation and experimental scenario were designed to verify the effectiveness of the algorithm.</div></div>","PeriodicalId":49856,"journal":{"name":"Mathematics and Computers in Simulation","volume":"240 ","pages":"Pages 571-588"},"PeriodicalIF":4.4,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144758066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reverse order law for NDMPI of dual complex matrices and its applications","authors":"Tikesh Verma, Amit Kumar, Debasisha Mishra","doi":"10.1080/03081087.2025.2537965","DOIUrl":"https://doi.org/10.1080/03081087.2025.2537965","url":null,"abstract":"","PeriodicalId":49905,"journal":{"name":"Linear & Multilinear Algebra","volume":"7 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144737009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A topological algorithm for the Fourier transform of Stokes data at infinity","authors":"Jean Douçot,&nbsp;Andreas Hohl","doi":"10.1112/jlms.70253","DOIUrl":"https://doi.org/10.1112/jlms.70253","url":null,"abstract":"<p>We give a topological description of the behaviour of Stokes matrices under the Fourier transform from infinity to infinity in a large number of cases of one level. This explicit, algorithmic statement is obtained by building on a recent result of T. Mochizuki about the Fourier transform of Stokes data of irregular connections on the Riemann sphere and by using the language of Stokes local systems due to P. Boalch. In particular, this induces explicit isomorphisms between wild character varieties, in a much larger range of examples than those for which such isomorphisms have previously been written down. We conjecture that these isomorphisms are compatible with the quasi-Hamiltonian structure on the wild character varieties.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"112 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70253","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144716703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}